Periodic total variation flow of non-divergence type in Rn

TL;DR

This paper introduces a new concept of viscosity solutions for highly singular nonlinear parabolic equations of non-divergence form, including total variation flow and crystalline mean curvature motion, in periodic domains.

Contribution

It develops a novel framework for viscosity solutions applicable to non-divergence type problems with nonlocal diffusion in arbitrary dimensions.

Findings

Established a comparison principle for the new solutions.

Proved stability under approximation by regularized problems.

Demonstrated existence for continuous initial data.

Abstract

We introduce a new notion of viscosity solutions for a class of very singular nonlinear parabolic problems of non-divergence form in a periodic domain of arbitrary dimension, whose diffusion on flat parts with zero slope is so strong that it becomes a nonlocal quantity. The problems include the classical total variation flow and a motion of a surface by a crystalline mean curvature. We establish a comparison principle, the stability under approximation by regularized parabolic problems, and an existence theorem for general continuous initial data.